Q239 / 270 · NDA 2015 (Phase 2) · Mathematics

$\int_{-1}^1 \mathrm{x}|\mathrm{x}| \mathrm{dx} \text { is equal to }$

  • Correct answer
    A
    0
  • B
    2/3
  • C
    2
  • D
    -1

Solution

$\begin{aligned} & \int_{-1}^1 x|x| d x \\ & =\int_{-1}^0 x|x| d x+\int_0^1 x|x| d x \\ & =\int_{-1}^0 x(-x) d x+\int_0^1 x \cdot x d x \\ & =-\int_{-1}^0 x^2 d x+\int_0^1 x^2 d x \\ & =-\left[\frac{x^3}{3}\right]_{-1}^0+\left[\frac{x^3}{3}\right]_0^1 \\ & =-\left[0-\frac{(-1)^3}{3}\right]+\frac{1}{3}\left[(1)^3-(0)^3\right] \\ & =-\frac{1}{3}+\frac{1}{3}=0\end{aligned}$

Hence, the correct answer is option: 'A'.

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